A Guide to Writing Mathematics
Dr. Kevin P. Lee
Introduction
This is a math class! Why are we writing?
There is a good chance that you have never written a paper in a math class before. So you
might be wondering why writing is required in your math class now.
The Greek word mathemas, from which we derive the word mathematics, embodies the
notions of knowledge, cognition, understanding, and perception. In the end, mathematics is
about ideas. In math classes at the university level, the ideas and concepts encountered are
more complex and sophisticated. The mathematics learned in college will include concepts
which cannot be expressed using just equations and formulas. Putting mathemas on paper
will require writing sentences and paragraphs in addition to the equations and formulas.
Mathematicians actually spend a great deal of time writing. If a mathematician wants
to contribute to the greater body of mathematical knowledge, she must be able
communicate her ideas in a way which is comprehensible to others. Thus, being able to
write clearly is as important a mathematical skill as being able to solve equations.
Mastering the ability to write clear mathematical explanations is important for
non-mathematicians as well. As you continue taking math courses in college, you will come
to know more mathematics than most other people. When you use your mathematical
knowledge in the future, you may be required to explain your thinking process to another
person (like your boss, a co-worker, or an elected official), and it will be quite likely that
this other person will know less math than you do. Learning how to communicate
mathematical ideas clearly can help you advance in your career.
You will find that writing good mathematical explanations will improve your knowledge
and understanding of the mathematical ideas you encounter. Putting an idea on paper
requires careful thought and attention. Hence, mathematics which is written clearly and
carefully is more likely to be correct. The process of writing will help you learn and retain
the concepts which you will be exploring in your math class.
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What does good mathematical writing look like?
As you learn more math, being able to express mathematical ideas will become more
important. It will no longer be sufficient just to be able to write down some final “answer”.
There is a good reason why Herman Melville wrote Moby Dick as a novel and not as the
single sentence:
The whale wins.
For this same reason, just writing down your final conclusions in an assignment will not be
sufficient for a university math class.
You should not confuse writing mathematics with “showing your work”. You will not
be writing math papers to demonstrate that you have done the homework. Rather, you
will be writing to demonstrate how well you understand mathematical ideas and concepts.
A list of calculations without any context or explanation demonstrates that you’ve spent
some time doing computations; however, a list of calculations without any explanations
omits ideas. The ideas are the mathematics. So a page of computations without any
writing or explanation contains no math.
When you write a paper in a math class, your goal will be to communicate
mathematical reasoning and ideas clearly to another person. The writing done in a math
class is very similar to the writing done for other classes. Your are probably already used
to writing papers in other subjects like psychology, history, and literature. You can follow
many of the same guidelines in a mathematics paper as you would in a paper written about
these other subjects.
Basics: Combining Words and Equations
Following the rules of grammar.
Good writing observes the rules of grammar. This applies to writing in mathematics
papers as well! When you write in a math class, you are expected to use correct grammar
and spelling. Your writing should be clear and professional. Do not use any irregular
abbreviations or shorthand forms which do not conform to standard writing conventions.
Mathematics is written with sentences in paragraphs. (And yes, paragraphs are important.
It is not amusing to read a three-page paper consisting of just one paragraph.)
There is however one element in mathematical writing which is not found in other types
of writing: formulas. However, it may surprise you to know that in a math paper, formulas
and equations follow the standard grammatical rules that apply to words. Mathematical
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symbols can correspond to different parts of speech. For instance, below is a perfectly good
complete sentence.
1 + 1 = 2.
The symbol “=” acts like a verb. Below are a couple more examples of complete sentences.
3xy < −2.
5z ∈ R.
9 − s 6= t.
Can you identify the verbs? On the other hand, an expression like
5x2 z − 10y
is not a complete sentence. There is no verb. Such an expression should be treated as a
noun. Can you identify the nouns in the previous examples?
Formulas and equations need to be contained in complete sentences with proper
punctuation. Here is an example:
The total revenue, R, made from selling widgets is given
by the equation
R = pq,
where p is the price at which each widget is sold and q is
the number of widgets sold. Based on past experience,
we know that when widgets are priced at $15 each, 2000
widgets will be sold. We also know that for every dollar
increase in price, 150 fewer widgets are sold. Hence, if
the price is increased by x dollars, then the revenue is
R = (15 + x)(2000 − 150x)
= −150x2 − 250x + 30, 000.
Notice how punctuation follows each of equations. A computation which ends a sentence
needs to end with a period. Computations which do not end sentences are followed by
commas.
A good way to improve your mathematical writing is by reading your writing, including
all of the equations, out loud. Your ears can often pick out sentence fragments and
grammatical errors better than your eyes. If you find yourself saying a series of fragmented
sentences and equations, you should do some rewriting.
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There are a couple of other important things to observe in the above example. Notice
how “we” is used. The use of first person is common in mathematics, especially the plural
“we”, so don’t be afraid to use the word “we” in the papers you write in your math class.
Another thing to notice is that important or long formulas are written on separate
lines. You can make your mathematical writing easier to read if you place each important
formula on a line of its own. It’s hard to pick out the important formulas below:
If d is Bob’s distance above the ground in feet, then d =
100 − 16t2 , where t is the number of seconds after Bob’s
Flugelputz-Levitator is activated. Solving for t in the
equation 100 − 16t2 = 0, we find that t = 2.5. Bob hits
the ground after 2.5 seconds.
This is clearer:
If d is Bob’s distance above the ground in feet, then
d = 100 − 16t2 ,
where t is the number of seconds after Bob’s
Flugelputz-Levitator is activated. Solving for t in the
equation
100 − 16t2 = 0,
we find that t = 2.5. Bob hits the ground after 2.5 seconds.
Symbols and words.
It is important to use words and symbols appropriately. Part of being able to write
mathematics well is knowing when to use symbols and knowing when to use words.
Don’t use mathematical symbols when you really mean something else. A common
mistake is to misuse the “=” symbol. For instance:
32x − 2(3x ) = −1 = (3x )2 − 2(3x ) + 1 = 0 =
!!
(3 − 1) = 0 = 3 = 1 = x = 0.
x
2
x
Do not use the equal sign when you really mean “the next step is” or “implies”. The above
example is really saying that −1 = 0 = 1! Using arrows instead of equal signs is a slight
improvement, but still not desirable:
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32x − 2(3x ) = −1 → (3x )2 − 2(3x ) + 1 = 0 →
!
(3x − 1)2 = 0 → 3x = 1 → x = 0.
With a sequence of calculations, sometimes it is best to just place each equation on a
separate line.
32x − 2(3x ) = −1
(3x )2 − 2(3x ) + 1 = 0
(3x − 1)2 = 0
3x = 1
x = 0.
For a difficult computation where the reader might not readily follow each step, you can
include words to describe the steps you take.
We want to solve for x in the equation
32x − 2(3x ) = −1.
We can rewrite this equation in terms of 3x :
(3x )2 − 2(3x ) + 1 = 0.
After factoring, this becomes
(3x − 1)2 = 1
and it follows that 3x = 1, or x = 0.
However, make sure that your paper has a single flow. Don’t explain a calculation using
the “two-column method”.
32x − 2(3x ) = −1 Solve this equation.
(3x )2 − 2(3x ) + 1 = 0
(3 − 1) = 0
x
Collect the terms on one side.
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Factor.
x
Use the Zero Factor Property.
x=0
Solve for x.
3 =1
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This is hard to read through. It’s also bad style.
Some things are best expressed with words. But other things are best expressed with
mathematical notation. For instance, it hard to read:
It follows that x plus two is larger than zero.
Here, mathematical notation is more appropriate.
It follows that x + 2 > 0.
Miscellaneous comments.
Here are a couple of other pointers to help you get started with your mathematical writing.
• Don’t start a sentence with a formula. While it may be grammatically correct, it
looks strange.
t = 5 when w = 2000, so we can conclude that the new
factory will be completely overrun with cockroaches in 5
years.
f is globberfluxible at x = 3.
Adding just a word or two can fix these examples.
Since t = 5 when w = 2000, we can conclude that the
new factory will be completely overrun with cockroaches
in 5 years.
The function f is globberfluxible at x = 3.
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• Don’t turn in pages of unreadable scribbles to your professor. In college, papers are
typed. They are also usually double-spaced with large margins. Mathematics papers
adhere to the same standards as papers written for other classes.
• While it is a good idea to type your paper, you may have to leave out the formulas
and insert them by hand later. It is perfectly acceptable to write formulas by hand in
a math paper. Just make sure that your mathematical notation is legible. If you do
decide to type the equations, please be aware that variables in equations and formulas
are usually italicized (to set them apart from the text). Many word processing
programs contain equation editors. In newer versions of Microsoft Word, the equation
editor is available under the Insert menu. Select Object..., and then Equation.1 If
you are going to be writing a lot of technical documents, it might be worthwhile to
learn TEX or LATEX. These are professional mathematical typesetting languages. This
document was written with LATEX. You may also find satisfactory results typing
papers in Maple or some other mathematically oriented software program.
• Use mathematical notation correctly. As you learn to write more complicated
formulas, it is all too easy to leave out symbols from formulas. Learn how to use
symbols properly!
• Use language precisely and correctly. Make sure that the words you use really mean
what you think they mean. Mathematics requires very precise use of language.
Another thing to avoid is overuse of the word “it”. Mathematical papers with a lot of
pronouns like “it” and “that” tend to be hard to read. It is often hard for the reader
to see what “it” is referring to. If you, the author, are also having difficulty seeing
what “it” is referring to, then you may be having some difficulty with the
mathematical ideas; you may need to think more about the ideas you are writing
about.
• Try to write as simply and directly as possible. No one likes to read ponderous
pretentious prose.
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In Microsoft Word, it is also possible to place a button on the tool bar which activates the equation
editor. Select Configure... beneath the Tools menu. In the window that pops up, select the Commands
tab. Under the Insert category you will find the Equation Editor command. Drag the equation editor
icon to the tool bar.
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Mathematical Ideas into Writing
Organizing your paper.
A well-organized paper is easier to read than a disorganized one. Fortunately, there are
some standard ways to order a mathematics essay.
First, there is some type of introduction. Usually, the introduction states the problem.
Even if you are answering a problem from a text book, you should not assume that the
reader is familiar with the text book or even has a copy of the text book available to him
or her. However, do not just copy the problem! You must rewrite the problem in your own
words.
A good introduction should also discuss the significance of the problem. The
introduction is where you will need to “hook” the reader.
It is not a bad idea to also preview the rest of the paper in the introduction. Give the
reader some idea of what to expect later.
We will analyze the revenue using a linear model and then
examining the graphs generated by the model.
The production of fava beans will be modeled using a C
program.
First, we will analyze the population using numerical
methods. Then, we will analyze the population using
formulas. We will then compare the two different results.
Some papers then state the “answer” to the problem right after the introduction. Other
papers place the “answer” at the end. This is a matter of taste. Sometimes, the end result
is the most important thing in the paper. You may need to place the end result at the
beginning to entice the reader. On the other hand, sometimes the method of arriving at
the end result is more important. In such a case, putting the result at the end may be
more sensible.
In any case, it is best to state the result in terms of the original problem using
real-world terms.
The solution is t = 6.
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The solution to the equation is t = 6. The population of
Utopia is at its smallest 6 years after the plague begins.
Make sure that the arguments you write are carefully organized. It may help you to
write an outline before you begin writing a mathematics paper. Writing an outline will also
help you think about the concepts more clearly and thus will help you learn the material.
As you write about more advanced mathematical problems, organization will become even
more important.
Writing for your audience.
For most papers that you write in your math class, you should assume that the reader has
about the same mathematical knowledge that you have. When you write up the solution to
a homework problem, it might be helpful to think that you are writing to a student in
another section of the same class or in a similar class at another school. Some of the papers
you will be writing will be directed toward a reader who may know less math. The purpose
of a math paper is not just to show the professor that you know something. Your math
professor already knows the subject; you are not writing for his or her benefit. You are
writing for someone who doesn’t know the subject. (That someone may be you! You can
use your writing assignments to help review for exams.)
In your mathematics writing, you will be communicating to the reader why and how
you arrived at a solution. You will also want to convince your reader that your particular
reasons and your particular means to the solution are correct. A good mathematical paper
not only should provide clear explanations, but should also be able to persuade a skeptical
reader.
Many times, if you can arrive at the same solution through alternate routes, you can
make your writing more persuasive. You may want to analyze a problem using both
computers and algebra. Or you might compare a graph with real-world information.
Pictures and graphical depictions can be very helpful for your reader.
Specific examples will also help to make your writing more persuasive. You can help a
reader understand an abstract general argument by showing how the argument applies to a
specific case. You can also use “extreme” cases to show the limits of an argument.
Make sure that what you write is relevant to the problem. Including extraneous
comments or information demonstrates a lack of understanding of the ideas and concepts,
and reduces the overall effectiveness of your mathematical writing. Thinking about the
reader will help you to decide which details you need to include and which details you
should leave out. Calculations which are tedious and uninteresting to the reader can be
readily omitted. (Again, mathematics writing is not the same as showing work. You don’t
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need to show everything.) The reader of a college mathematics paper will probably not be
interested in reading how to multiply 5 and 74. Leave out what is unimportant. On the
other hand, don’t leave out anything which is critical to the key ideas you are trying to
explain. Learning what is important and what is unimportant will help you understand
mathematics better.
You should not assume that the reader is familiar with the problem you are solving.
While you do not need to restate the problem in its entirety, be sure to give an overview of
all important details in the problem. You also should not assume that the reader is in the
same mind set as you. In your writing, state any assumptions which you have made. For
instance, in physics problems, it is often assumed that everything is frictionless. But just
because this assumption is made nearly all the time doesn’t mean that your reader will
automatically make this assumption; your reader may not be familiar with physics. Just
because you assume something is true doesn’t mean that your reader will. So write it down!
Defining variables and formulas.
Quantities and functions can be, and often should be, represented with letters. However,
the letters which are chosen are arbitrary. You should explicitly state what all letters in
your formulas represent in as precise a manner as possible. For instance:
Either n or n + 1 is even.
What is n? If n = 8.5 is the above statement true? A better way of stating this is:
For any whole number n, either n or n + 1 is even.
A common phrase used in mathematics is “Let...”.
Let x be any real number.
Let P be the population of Los Angeles in 2010.
Let f (x) = x2 + 1.
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In the last example, x is a place holder. It doesn’t require a proper introduction. However,
it would be better to write:
Let f (x) = x2 + 1 for all real numbers x.
If describing all the variables gets tedious, try not assigning any variables at all. The
following example clearly needs improvement.
The volume is `wh.
The following example is adequate, but wordy.
The volume of the box is `wh, where ` is the length, w is
the width, and h is the height.
We can write this most elegantly by removing the variables.
The volume of the box is the product of the length, the
width, and the height.
!
You need to be especially careful with variables representing real-world quantities.
Avoid describing them vaguely, as in:
Let D(t) be the distance at a time t.
Including units would make this clearer, but the description is still vague.
Let D(t) be the distance in miles at t hours.
Try to be as specific as possible.
Let D(t) be Agnes’s distance from the arena in miles t
hours after the riot began.
Also, be careful that each symbol you use represents only one thing. This can actually
be more subtle than it sounds. The following example seems to be rather clear.
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Let P be the escaped wombat population (in thousands)
t years after 1990 and suppose that
P = 0.5(1.12)t .
The wombat population in 1992 is approximately 672.
We can see this by setting t = 2 and observing that
P = 0.5(1.12)2 = 0.6272 thousand wombats.
If we want to predict when the wombat population will
reach 2000, we set P = 2 and solve for t using
logarithms.
2 = 0.5(1.12)t
log 2 = log 0.5 + t log 1.12
log 2 − log 0.5
t=
≈ 12.23 years.
log 1.12
The wombat population will reach 2000 in the year 2002.
I think that the above example would be considered unobjectionable by most readers. It
looks very clear and understandable. The variable P is always standing for the wombat
population. However, notice that in the first paragraph, P is the wombat population in
general. In the next paragraph, P = 0.6272, the wombat population in 1992. And in the
last paragraph, P = 2. The meaning of P appears to be changing every time that it is
used. In the first paragraph, P represents the population at any time. In the other
instances, P represents the population at one particular time. The problem can be fixed
omitting some variables and adding others.
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Let P be the escaped wombat population (in thousands)
t years after 1990 and suppose that
P = 0.5(1.12)t .
By substituting 2 for t in the above equation, we can see
that in 1992, the wombat population is approximately
672.
0.5(1.12)2 = 0.6272 thousand wombats.
Let t2000 be the year when the wombat population
reaches 2000. Then,
2 = 0.5(1.12)t2000
log 2 = log 0.5 + t2000 log 1.12
log 2 − log 0.5
t2000 =
≈ 12.23 years.
log 1.12
The wombat population will reach 2000 in the year 2002.
While in the above example, we can afford a little bit of sloppiness with the variables, in
more complex problems, this could be a source of potential trouble. When a symbol is used
to represent two different things (even, or perhaps especially, if those things are similar),
the reader (and the writer!) can become confused. A symbol used in two different ways is
not only confusing, but often results in incorrect mathematics!
Just as variables need to be introduced carefully, also be sure not to pull formulas out
of thin air. Tell the reader how you get each formula or what each formula means. It’s not
very pleasant to get hit with formulas without any warning.
Using pictures in mathematics.
A picture can really be worth a thousand words. I strongly encourage you to use visual
arguments in your mathematical writing. However, if you do include a picture, a diagram,
a graph, or some other visual mathematical representation, make sure that you fully
explain how it fits into your mathematical argument.
Looking at the graph, we can see that the result is true.
What should the reader look for in the graph? Why does the graph support the argument?
Be more specific.
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The graph increases sharply at t = 3, confirming our
earlier prediction that the robots will begin a homicidal
rampage three years from now.
A good graph should convey relevant and specific information to the reader. The
following graph is vague.
Graphs and diagrams need to be neatly drawn and clearly labeled. Indicate the scale on
the axes. You should point out significant graphical features.
Cooties infections versus time
No. of infections I (in thousands)
1000
•
maximum number of infections
900
800
700
600
500
400
300
200
100
10 20 30 40 50 60 70 80 90 100
Time t after epidemic begins (in days)
If you draw a graph by hand, use a straight edge. You may want to generate your graphs
using a computer. Be careful though. Programs like Excel or Microsoft Office generally are
not good at generating mathematical graphs. You will more likely have success using a
math program like Maple.
Any diagrams you draw should also be carefully labeled. Be sure to label everything
that you refer to in your argument.
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Epilogue
Writing mathematics is not the easiest thing to do. Writing mathematics is a skill which
takes practice and experience to learn. There are many resources here at Purdue Calumet
which are available to you to help you with your mathematical writing. Among these are
the Math Lab and the Writing Lab.
If you have not written mathematics much before, it may feel frustrating at first. But
learning to write mathematics can only be done by actually doing it. It may be hard at
first, but it will get easier with time and you will get better at it. Do not get discouraged!
Being able to write mathematics well is a good skill to learn, and one which you will keep
for a lifetime.
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A mathematical writing checklist
Below is a checklist which will help you follow the guidelines outlined above in your
mathematical writing.
Is your paper neatly typed?
If you write the equations by hand, make sure that you have written in all of the
equations. Also make sure that you have included all of the diagrams and graphs you
intended to. Make sure that the paper is double-spaced and has wide enough margins.
Has the paper been proofread?
In college, sloppy work is not appreciated. Do check over everything.
Is there an introduction?
Make sure that you explain the problem to the reader. Assume that the reader is
unfamiliar with the problem. The introduction should also try to indicate to the
reader why the problem is interesting and give some indication of what will follow in
the paper.
Did you state all of your assumptions?
Write down any physical assumptions that you made. (Did you assume that there was
no friction? That the population grew with unlimited resources? That interest rates
remained steady?) Write down any mathematical assumptions that you made. (Did
you assume that the function was continuous? Linear? That x was a real number?)
Are the grammar, spelling, and punctuation correct? Is the writing clear
and easy to understand?
Make sure that there are no sentence fragments. The formulas and equations too
need to be contained in complete sentences. Equations and formulas (and the words
too) should have correct punctuation as well. Make sure that your paper flows
smoothly and reads well. And please, don’t be careless! Check your spelling!
Are all of the variables defined and described adequately?
Make sure that you introduce each variable that you use. Describe each variable as
precisely as possible. Don’t forget any units!
Are the mathematical symbols used correctly?
Don’t use an “=” sign outside of a formula. Make sure that the symbols are not
misused. Use equations and formulas where they are appropriate.
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Are the words used correctly and precisely?
Avoid using vague language and too many pronouns. Use words where they are
appropriate.
Are the diagrams, tables, graphs, and any other pictures you include
clearly labeled?
Graphs should be drawn with a straight edge (or computer-generated) with axes
clearly labeled (with units if appropriate) and the scale indicated. Diagrams should
be neatly drawn with relevant labels.
Is the mathematics correct?
This should be obvious.
Did you solve the problem?
Sometimes in all of the fuss, people forget to answer the problem. Do answer the
question! Also, see if you can write the solution in “real-world” terms.
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